Simulation applications to support teaching and research in epidemiological dynamics

Background An understanding of epidemiological dynamics, once confined to mathematical epidemiologists and applied mathematicians, can be disseminated to a non-mathematical community of health care professionals and applied biologists through simple-to-use simulation applications. We used Numerus Model Builder RAMP Ⓡ (Runtime Alterable Model Platform) technology, to construct deterministic and stochastic versions of compartmental SIR (Susceptible, Infectious, Recovered with immunity) models as simple-to-use, freely available, epidemic simulation application programs. Results We take the reader through simulations used to demonstrate the following concepts: 1) disease prevalence curves of unmitigated outbreaks have a single peak and result in epidemics that ‘burn’ through the population to become extinguished when the proportion of the susceptible population drops below a critical level; 2) if immunity in recovered individuals wanes sufficiently fast then the disease persists indefinitely as an endemic state, with possible dampening oscillations following the initial outbreak phase; 3) the steepness and initial peak of the prevalence curve are influenced by the basic reproductive value R0, which must exceed 1 for an epidemic to occur; 4) the probability that a single infectious individual in a closed population (i.e. no migration) gives rise to an epidemic increases with the value of R0>1; 5) behavior that adaptively decreases the contact rate among individuals with increasing prevalence has major effects on the prevalence curve including dramatic flattening of the prevalence curve along with the generation of multiple prevalence peaks; 6) the impacts of treatment are complicated to model because they effect multiple processes including transmission, recovery and mortality; 7) the impacts of vaccination policies, constrained by a fixed number of vaccination regimens and by the rate and timing of delivery, are crucially important to maximizing the ability of vaccination programs to reduce mortality. Conclusion Our presentation makes transparent the key assumptions underlying SIR epidemic models. Our RAMP simulators are meant to augment rather than replace classroom material when teaching epidemiological dynamics. They are sufficiently versatile to be used by students to address a range of research questions for term papers and even dissertations. Supplementary Information The online version contains supplementary material available at (10.1186/s12909-022-03674-3).


Exercise 2.
Plot (either as a series of 1-D curves or a 2-D surface) the relationship between the ratio of peak to endemic prevalence as a function of 0 (through manipulation of the value 0 ) and the mean residence timēR = 1∕ RS Exercise 3.
By varying the strength of adaptive behavior through changes in the prevalence values of the behavioral switch parameter estimate the period of endemic oscillations that dampen over a five-year simulation interval as a function of the values of and the mean waiting timēR = 1∕ RS for combinations of these parameters where dampened oscillatory behavior is evident. (Note: this will require identification of local maxima in the prevalence curve over the five year simulation interval and computing the average time between consecutive local maxima over the five year period for each of selected pair of parameter values.) Come up with creative ways to explore the complexities of including treatment as a consequence of the rate at which individuals are treated, limitations on the number of individuals that can be treated at anyone time, and assumed effects of treatment. These effects include reducing mortality in the population as a whole and making assumptions about mortality rates for individuals that are treated.

Exercise 6.
Come up with creative ways to explore the complexities of vaccination programs as a consequence of starting dates, vaccination rates, and limitations on the number of regimens available-either in absolute times or in monthly tranches.

Solution to Exercise 4.1
In Figure A.1 (left panel) we report the histograms of times at which the prevalence ( ) became 0, for values of 0 varying from 1 to 8 and with other parameter value fixed as in the stochastic SIRS RAMP depicted in Fig. 3. In Figure  A.1 (right panel) we show the outbreak probability values plotted in red when computed from the model parameters using Eq. 22 (to compute 0 ) and Eq. 25 and plotted in blue when calculated using the proportion of simulations for which we observed a major outbreak, as shown in Figure 13. On the left we show the histograms of times at which the prevalence ( ) became 0, as a function of the value of the parameter value 0 . On the right panel, we show the estimation of the outbreak probability using two different approaches for its calculation (noting for the parameters used in our simulation that 0 = 1.2 0 ).

B. Boxcar Models
The exponential transfer distribution of Eq. 1.1 in Box 1 implies a maximum transfer rate of 1∕ at = 0 of a flow through disease class X. This is unrealistic since one would expect the maximum transfer rate not to be the moment of entering but rather around the some mean period of time in X. This can be remedied by subdividing X into sub-compartments x , = 1, ..., each of which is traversed at a rate . In this case we obtain a boxcar transfer process through disease class X, modeled by the following system of equations The solution X(t) to this systems of equation is known to be 1 minus the Erlang distribution multiplied by 0 . Specifically, This also implies in the context of individuals passing through X-that is, through all boxcars that constitute X-that with corresponding Erlang probability density function (shape parameter , scale parameter Thus, by computing ∫ The mode is no longer at 0, but now is somewhat below the mean with a value −1 . The variance of this distribution is 1 2 and as → ∞, all individuals spend the same amount of timē= 1∕ in X.

C. Waiting Times in SIRS ABMs
Suppose the per capita outflow rate of individuals from disease class is an increasing function of how long these individuals have been in X. We investigate the consequences of such an assumption by considering the exit distribution of 0 individuals who entered disease class X at time 0 under the assumption that their per capita rate of outflow from X is given by the function ( ) = 0 , for constants 0 > 0 and ≥ 0. This process can be described by the differential equation (compare with equations in Box 1) This also implies that with corresponding probability density function Thus, by computing ∫